Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
In Finite Monoid Computations and a Uniform Generalized Star-Height Bound
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every finite alphabet \Sigma and every regular language L\subseteq\Sigma^*,
In Generalized Star Height at Most Four
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every finite alphabet \Sigma and every regular language L\subseteq\Sigma^*, one has h_\Sigma(L)\le4.
No Lean statement in the release is paired with this theorem.
In Generalized Star Height at Most Three
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every finite alphabet \Sigma and every regular language L\subseteq\Sigma^*,
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
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Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
GeneralizedStarHeightThe paper states
In Finite Monoid Computations and a Uniform Generalized Star-Height Bound
For every finite alphabet \Sigma and every regular language L\subseteq\Sigma^*,
The words they share: regular, finite, alphabet, language, star, height.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
GeneralizedStarHeight.lean, the release lists it as a supporting statement.
theorem main {Alphabet : Type u} [Finite Alphabet] (L : Language Alphabet) (hL : L.IsRegular) : HasHeightAtMost L 3 := by sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
HasHeightAtMost Model.lean, line 39MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note: None.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
Generalized star height measures the nesting of Kleene stars in regular expressions that also allow Boolean operations. The formalization proves that every regular language over a finite alphabet has a generalized expression over the same alphabet of star height at most three. This is stronger than the bound of thirteen in the accompanying finite-monoid paper. The selected statement asserts the uniform expression bound, without separately encoding every construction step of that paper.
The formalization proves that every regular language over a finite alphabet has a generalized regular expression over that same alphabet with star height at most three. Generalized expressions allow Boolean operations as well as concatenation and Kleene star. The bound of three implies the accompanying paper’s bound of four; the selected statement concerns the expression bound itself. Generalized star height measures the nesting of Kleene stars in regular expressions that also allow Boolean operations. The formalization proves that every regular language over a finite alphabet has a generalized regular expression over that same alphabet of star height at most three. Complements are taken in the same free monoid. This is the paper’s uniform bound.
Uniform generalized star-height bound of three GeneralizedStarHeight
Unreviewed AI draft: states a paper’s theorem as the paper does. States the bound of three, the main theorem of the newest accompanying paper; it implies the bounds of four and thirteen that are the main theorems of the other two accompanying papers.
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
GeneralizedStarHeightThree reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 123 s; peak memory 6.3 GiB). | 11 October 2026 | Original report |
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
Author not stated. Source
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Why this passage is shown
Theorem 1.1
In Finite Monoid Computations and a Uniform Generalized Star-Height Bound
Why this passage is shown
Theorem 1.1
In Generalized Star Height at Most Four
Why this passage is shown
Theorem 1.1
In Generalized Star Height at Most Three
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, GeneralizedStarHeight
Proof-verification report
MathVet, GeneralizedStarHeight
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, GeneralizedStarHeight
Lean statement
GeneralizedStarHeight